My husband's former spouse, who has an agreement for lifetime spousal support after a marriage of around eight years, and also receives child support, is destroying my husband. He's depressed, medicated, and our finances are always subject to inspection. He works constantly. Is there any precedent for new spouses bringing suit against former spouses?
There's no established legal basis for making a claim like this. In fact, your claim would be so new and different that you'd risk being slapped with a penalty for filing what's called a "frivolous" suit, meaning that it's without legal grounds.
Your real object here seems to be relieving the burden on your husband, and a better way to go about that would be to try revisiting that lifetime spousal support obligation. An obligation like this is unusual for a marriage of less than ten years. If this obligation was adopted by the court as part of your husband's divorce, he may be stuck with it. But if it wasn't, an experienced family lawyer might be able to help your husband get out of the obligations. And, if you can prove that your husband is doing as poorly as you describe, a sympathetic judge might lower his payments. For this, you would need to file a motion with the court.
The child support is a different matter. Once someone brings a child into this world, they become legally responsible for supporting that child until the child becomes an adult. Although your husband might be able to get a downward adjustment in child support if he can really show changed circumstances (like a serious illness that prevents him from working), he should first consider carefully what's best for his child.